In order to frame our results in the context of the currently existing elliptic theory, let us review some historical milestones. In the middle of the 20th century the theory of boundary value problems mainly concentrated on the case when coefficients of the underlying equations and domains exhibit some amount of smoothness. The past 30-40 years have brought great developments in the study of elliptic measure and boundary value problems for operators with non-smooth bounded measurable coefficients. The background theory of weak solutions, Green function estimates, maximum principle, and similar results were extended to all divergence form elliptic operators with bounded measurable coefficients. It turned out, however, that the question of absolute continuity of the resulting elliptic measure with respect to the Lebesgue measure on the boundary, or, equivalently, of well-posedness of the Dirichlet boundary value problem with boundary data in \(L^p\) , is much more delicate. First of all, examples have been found that show such results can not be expected for all elliptic operators and some regularity of the coefficients in the transversal direction to the boundary is, in fact, necessary [1]}, [2]}. In light of these examples, the initial efforts concentrated on the study of operators whose coefficients are constant in the transverse direction to the boundary. Later results have extended the theory to the optimal regularity of the coefficients, expressed in terms of a Carleson measure condition. In this survey, and in this paper, we shall concentrate on the fundamental case where the domain is the upper half-space \(\mathbb {R}_+^{n+1}=\left\lbrace (x,t)\in \mathbb {R}^n\times (0,\infty )\right\rbrace \)  and the coefficients of the operator are independent of the transverse direction, that is, \(t\) -independent. The first breakthrough in this direction was the 1981 paper of Jerison and Kenig [3]} which established well-posedness of the Dirichlet problem and the absolute continuity of the elliptic measure for operators with symmetric bounded measurable \(t\) -independent coefficients on \(\mathbb {R}^{n+1}_+\)  and, by a change of variables, above a graph of a Lipschitz function. A seemingly innocent assumption of symmetry turned out to be critical and it took 20 years to extend these results to non-symmetric operators in dimension 2 [4]} and more than 30 years to non-symmetric operators in any dimension [5]}. The 1981 work of Jerison and Kenig relied on the beautiful and powerful Rellich identity which roughly speaking says that the \(L^2\)  norms of the normal and tangential derivatives of solutions on the boundary are comparable. It is proved by an integration by parts argument invoking the symmetry of the coefficients. However, not only the method of the proof of the Rellich identity, but the \(L^2\)  equivalence of the norms of the normal and tangential trace of the solution itself fails when the coefficients are not necessarily symmetric. This has been demonstrated in [4]}, where the authors established extremely useful characterizations of solvability of the Dirichlet problem in \(L^p\)  in terms of the square function/non-tangential maximal function estimates (in any dimension), a method that made possible many later developments including the present paper, and resolved the question of absolute continuity of elliptic measure with respect to the Lebesgue measure for \(t\) -independent non-symmetric operators in dimension 2. Unfortunately, many ingredients in the argument in [4]} rely heavily on the space being 2 dimensional. For example, the 2-d case relies on a change of variable argument that does not carry forward to higher dimensions. Only 15 years later these results have been finally extended to multidimensional setting. In [5]} the authors established the square function/non-tangential maximal function estimates for solutions to the \(t\) -independent, non necessarily symmetric, operators on \(\mathbb {R}^{n+1}_+\)  for all \(n\) , and as a result, absolute continuity of the elliptic measure with respect to the Lebesgue measure and well-posedness of the Dirichlet boundary value problem in \(L^p\) . The method involved a new pull-back/push-forward sequence based on the Hodge decomposition of the coefficients, the celebrated solution to the Kato problem [9]}, the square function/non-tangential maximal function estimates for the heat semigroup, and many other elements. The method has later been streamlined in [10]} to avoid an explicit pull-back/push-forward on Lipschitz domain – an important development in our context.
